   (i) \(\forall \,0<s<t\) , by the Hölder inequality, one gets:
\(\begin{array}{rll}t-s&=\int _{s}^{t}\big (\frac{d}{dr}\int _{B(x_0;r)}(f^2+\epsilon )^{\frac{q}{2}}dv\big )^{\frac{1}{p}}\big (\frac{1}{\frac{d}{dr}\int _{B(x_0;r)}(f^2+\epsilon )^{\frac{q}{2}}dv}\big )^{\frac{1}{p}}dr\\&\le \bigg (\int _{s}^{t}\big (\frac{d}{dr}\int _{B(x_0;r)}(f^2+\epsilon )^{\frac{q}{2}}dv\big )dr\bigg )^{\frac{1}{p}}\bigg (\int _{s}^{t}\big (\frac{1}{\frac{d}{dr}\int _{B(x_0;r)}(f^2+\epsilon )^{\frac{q}{2}}dv}\big )^{\frac{1}{p-1}}dr\bigg )^{\frac{p-1}{p}}.\\\end{array}\) 
Taking this to the power of \(\frac{p}{p-1}\)  on both sides, we have
\(\begin{array}{rll}\bigg (\frac{(t-s)^p}{\int _{B(x_0;t)\backslash B(x_0;s)}(f^2+\epsilon )^{\frac{q}{2}}dv }\bigg )^{\frac{1}{p-1}}\le \int ^t_s\big (\frac{1}{\frac{d}{dr}\int _{B(x_0;r)}(f^2+\epsilon )^{\frac{q}{2}}}\big )^\frac{1}{p-1}dr\, .\end{array} \) 
Hence, for every strictly increasing sequence of \(\lbrace r_j\rbrace \)  and every
\(r_{\ell _0}>a\) , setting \(s=r_i, t=r_{i+1}\)  and summing over \(i\)  from
\(\ell _0\)  to \(\ell \)  and letting \(\ell \rightarrow \infty \) , one gets:
\(\begin{array}{rll}\sum \limits ^{\infty }_{i=\ell _0}\bigg (\frac{(r_{i+1}-r_{i})^p}{\int _{B(x_0;r_{i+1})\backslash B(x_0;s)}(f^2+\epsilon )^{\frac{q}{2}}dv}\bigg )^\frac{1}{p-1}\le \int ^{\infty }_{a}\big (\frac{1}{\frac{d}{dr}\int _{B(x_0;r)}|f|^q}\big )^\frac{1}{p-1}dr\, .\end{array} \) 
Let \(\epsilon \rightarrow 0\) , we get the desired first assertion by the
coarea formula and the Dominated Convergence Theorem. The proof of
the second assertion follows from Theorem REF  and Theorem .
(ii) By the convexity assumption, comparing the slope of the tangent and the slope of the chord at two distinct points (analogous to [1]}), we have
\(r > 0\, \Rightarrow \frac{d}{dr}\int _{B(x_0;r)}|f|^{q}dv\ge \frac{\int _{B(x_0;r)}|f|^{q}dv - \int _{B(x_0;0)}|f|^{q}dv}{r-0} \, \operatorname{a.e.}\, .\) 
Hence, \(\int _{a}^{\infty }\big (\frac{1}{\frac{d}{dr}\int _{B(x_0;r)}|f|^{q}dv}\big )^{\frac{1}{p-1}}dr\le \int _{a}^{\infty }\big (\frac{r}{\int _{B(x_0;r)}|f|^{q}dv}\big )^{\frac{1}{p-1}}dr< \infty \) . The assertion follows from the coarea formula.
